A Mixed Element Algorithm Based on the Modified L1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model

Fractal and Fractional

Zhichao Fang¹  Hong Li¹  Baoli Yin¹  Yang Liu¹  Jinfeng Wang² 

[1] School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China;School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China;

关键词: fourth-order fractional diffusion-wave equation;    modified L1-formula;    mixed element method;    a priori error estimates;   

DOI  :  10.3390/fractalfract5040274

来源: DOAJ

【 摘 要 】

In this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupled system with three equations is constructed. Compared to the previous space–time high-order model, the derived system is a lower coupled equation with lower time derivatives and second-order space derivatives, which can be approximated by using many time discrete schemes. Here, the second-order Crank–Nicolson scheme with the modified $L1$-formula is used to approximate the time direction, while the space direction is approximated by the new MFE method. Analyses of the stability and optimal $L^2$ error estimates are performed and the feasibility is validated by the calculated data.

【 授权许可】

Unknown